Finite-Dimensional Subalgebras of the Lie Algebra of Vector Fields on the Circle

Finite-dimensional subalgebras of the Lie algebra $\mathrm {Vect}(S^1)$ of smooth tangent vector fields on the circle are considered that consist of analytic vector fields. It is proved that (up to an isomorphism) there are only three such subalgebras: a one-dimensional subalgebra, a two-dimensional noncommutative subalgebra, and a three-dimensional subalgebra isomorphic to $\mathrm {sl}_2(\mathbb R)$.